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Journal ofApplied

Mechanics Brief NotesA Brief Note is a short paper that presents a specific solution of technical interest in mechanics butwhich does not necessarily contain new general methods or results. A Brief Note should not exceed1500 words or equivalenta typical one-column figure or table is equivalent to 250 words; a one lineequation to 30 words. Brief Notes will be subject to the usual review procedures prior to

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On the Relation Between the

L-Integral and the Bueckner

Work-Conjugate Integral

J. P. Shi, X. H. Liu, and J. M. LiDepartment of Engineering Mechanics, Xian University

of Technology, 710048 Xian, China

A simple but inherent relation between the L-integral and theBueckner work conjugate integral is deduced for crack problem inisotropic, anisotropic, and dissimilar materials, respectively. It is

proved the L-integral, from the mathematical point of view as wellas in principle, is arising from the Bettis reciprocal theorem.S0021-89360000103-3

1 Introduction

Knowles and Sternberg 1 have shown that the L-integral isgiven by

L

e3i jWx jn iTiujTkuk,ixjds (1)

where is a closed contour in the x 1x , x2y plane surroundinga whole crack; Wis the strain energy density, and Ti is the tractionacting on the outer side of the . The characteristics of the

L-integral and the Jk-integral are different. It can be proven thatthe L-integral is a path-independent integral. We can also verifythat the L-integral is independent of the selection of the coordinatesystem.

The Bueckner work-conjugate integral 2 was derived from

the well-known Bettis reciprocal theorem, which could be formu-lated as follows:

I

u iIi j

IIu i

IIi jInjds i,jx ,y (2)

where the superscripts I and II refer to two possibledisplacement-stress fields which satisfy the traction-free condi-

tions on the crack faces. The property of the path-independentintegral is proved by Bueckner using Bettis work reciprocal theo-rem, that is IIC .

2 ProofFor homogeneous isotropic materials, assume that the first pos-

sible displacement-stress field is induced by the following com-plex potentials (z) and (z):

z1

z2a 2k1

Ekzk

k1

Fkzk1

(3)

z1

z2a 2k1

Ekzk

k1

Fkzk1

where Ek and Fk are complex coefficients which can be definedby remote conditions. Introduce a supplemental displacement-

stress field defined by the following complex potentials (II)(z)

and

(II)

(z):IIz izz

(4)

IIz izz2 izz .

The corresponding displacement and stress components are de-rived as follows:

u iIIy u i ,xxu i ,y

i jIIyi j ,xxi j ,y

1

2 i j ,xdy 12 i j ,ydx i,j1,2

(5)

where u i and i j , as the I field, are the displacement and stresscomponents induced by Eq. 3. It can be examined that the stress

i jII satisfy the traction-free conditions on the crack faces. Substi-

tuting Eq. 5 and u i and i j into Eq. 2, we obtain

I u ii j,xyu ii j ,yxu i,xi jyu i,yi jx

1

2u i i j ,xdy 12 u i i j ,ydx njds (6)

where dxn 2ds , dyn1ds .Now, the I 2L is examined. Utilizing the equilibrium condi-

tions in plane problems and noting the integral terms Tiu i , i ju iand u ii j,xdy have no contribution for I2L when 0 atthe near of crack tip. Thus, I 2L is equal to zero. We obtain

Contributed by the Applied Mechanics Division of T HE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED

MECHANICS. Manuscript received by the ASME Applied Mechanics Division, June

6, 1999; final revision, January 21, 2000. Associate Technical Editor: J. T. Ju.

828 Vol. 67, DECEMBER 2000 Copyright 2000 by ASME Transactions of the ASME


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I2L. (7)

Equation 7 shows that between the L-integral and the Buecknerwork conjugate integral there is a simple but inherent relationship.We need not to know the obvious function expressions of com-plex potentials for the crack beforehand, but the traction-free con-ditions must be satisfied.

If there are two displacement-stress fields, namely, u i , i j and

u i(II)

, i j(II)

, using the Bettis reciprocal theorem to the region

bounded by crack borders, one can divide the contour C into CL ,CR and C ,C , where CL ,CR are circles around the left and the

right crack tips and C ,C are the straight line along the upperand lower crack faces, respectively. Because the stresses are freeon the crack faces, then IICLICR . In this case, if integrals

ICL and ICR can be evaluated for some displacement-stress fields,then the path-independent integral I can also be defined andevaluated. If the displacement-stress fields are defined by Eqs. 3and 4, we can deduce the following relationship between the

L-integral and the stress intensity factors:

L31 a

4K1LK2LK1RK2R (8)

where KLK1LiK2L , KRK1RiK2R are stress intensity fac-tors at the left and the right crack tips, respectively. and areelastic constants.

3 Discussion

The complex potentials of the center crack, 1(z),1(z),2(z) ,2(z), were obtained by Chen and Shi 3 by using thesame method obtained the eigenfunction expansion form by Rice4 in interfacial cracks for dissimilar material. The stress anddisplacement fields that are obtained from these complex poten-tials satisfy the traction-free conditions on the crack faces and thecontinuous condition along the entire interface.

A supplemental displacement-stress field defined by the

complex potentials 1(II)

(z),1(II)

(z),2(II)

(z) ,2(II)

is intro-

duced. The relations between 1(II)

(z),1(II)

(z),2(II)

(z),2(II)

and 1(z),1(z),2(z) ,2(z) are analogous to Eq. 4.In a similar manner, the displacement and stress of the II field

are presented in Eq. 5. They satisfy the traction-free conditionson the crack faces also. The corresponding displacement andstress components will be substituted into Eq. 6. Note that thecurve can be divided into two sections: curve 1 of the upperplane and the curve 2 of the below plane. The deductions of Eqs.6 to 7 relate to the equilibrium equations in plane problemsand traction-free conditions only, but dont involve the materialparameter. The process is the same as the above homogeneousisotropic material. Finally, we still obtain Eq. 7 in the interfacecrack, that is I2L.

Between L-integral and stress intensity factors there is the fol-lowing relation:

L 111

21

2 3K1LK2LK1RK2R

8 cosh2a (9)

where is oscillation index and KLK1LiK2L , KRK1RiK2R are complex stress intensity factors at the left and the rightcrack-tips, respectively. They cannot be separated into the pure Imodel and II model; 1 ,1 and 2 ,2 stand for the materialparameters of the upper and lower plane.

For anisotropy material, the Lekhenitski complex potentialtheory needs to be used 5. According to the need of the Bueck-ner work conjugate integral, the subsidiary stress-displacementfields, which represents II field, are

IIz1 iz 1z1

IIz2 iz 2z22iz2z1. (10)

The stress fields caused by Eq. 10 satisfy the traction-freeconditions. The relation between the II field and a physical stressfield are analogous to Eq. 5.

By proceeding in the same manner as the isotropic case fromEq. 6 to Eq. 7, we draw a conclusion I2L.

It can be seen that a simple but inherent relation between theL-integral and the Bueckner work conjugate integral is right allalong, although the characteristic of material is more complexthan isotropic and the complex potentials in these two cases aremore different with in isotropic.

4 ConclusionsUsing the Bueckner work conjugate integral through introduc-

ing a special subsidiary stress-displacement field, one can renderthe L-integral. The relation between L-integral and the Buecknerwork conjugate integral seems independent of the stress oscilla-tory singularities on the interface crack tips and the eigenroot inthe anisotropy. It is found that the L-integral, from the mathemati-cal point of view as well as in principle, is arising from the Bettisreciprocal theorem. This means that the Bueckner work conjugateintegral is a more general path-independent integral than othersare. Using the Bueckner integral through choosing a different sub-sidiary stress-displacement field could render any other path-independent integrals.

References1 Knowles, J. K., and Sternberg, E., 1972, On a Class of Conservation Laws in

Linearized and Finite Elastostatics, Arch. Ration. Mech. Anal., 44, pp. 187211.

2 Bueckner, H. F., 1973, Mechanics of Fracture: Methods of Analysis and So-lution of Crack Problems, G. C. Sih, ed., Noordhoff, Leyden, pp. 239314.

3 Chen, Y. H., and Shi, J. P., 1998, On the Relation Between the M-integraland the Bueckner Work-Conjugate Integral, Acta Mech. Sin., 30, pp. 495

502 in Chinese.4 Rice, J. R., 1988, Elastic Fracture Mechanics Concepts for Interface

Cracks, J. Appl. Mech., 55, pp. 98103.

5 Sih, G. C., and Chen, E. P., 1981, Cracks in Composite Materials, Mech.Fracture, 6, pp. 199.

A Note on the Driving Traction Acting

on a Propagating Interface:Adiabatic and Non-Adiabatic Processes

of a Continuum

R. AbeyaratneMem. ASME, Department of Mechanical Engineering,

Massachusetts Institute of Technology, Cambridge,

MA 02139

J. K. KnowlesFellow ASME, Division of Engineering and Applied

Sciences, California Institute of Technology, Pasadena,

CA 91125

An expression for the driving traction on an interface is derived for an arbitrary continuum undergoing an arbitrary thermome-chanical process which may or may not be adiabatic.S0021-89360000403-7



MECHANICS. Manuscript received by the ASME Applied Mechanics Division, July

30, 1997; final revision, Aug. 5, 1997. Associate Technical Editor: L. T. Wheeler.

Copyright 2000 by ASMEJournal of Applied Mechanics DECEMBER 2000, Vol. 67 829


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1 Summary

In this note we derive an expression for the driving traction,or Eshelby force 1, acting on a propagating interface in a con-tinuum. The interfaces that we have in mind might represent, forexample, a shock wave or a boundary between two phases of amaterial, and the thermomechanical processes which the con-tinuum is permitted to undergo may or may not be adiabatic. Fromthe perspective of irreversible thermodynamics, the driving trac-tion corresponds to a thermodynamic affinity; see, for ex-ample, 2 4. It plays a central role in modeling the kinetics ofphase transformations by characterizing the rate of propagation of

phase boundaries e.g., see 5 8.The derivation sketched below makes no assumptions about the

constitutive law for the continuum under consideration. Whenspecialized to a thermoelastic material, the expression for the driv-ing traction obtained here has certain similarities with the Leg-endre transform of the Helmholtz free-energy F, with respectto both the deformation gradient tensor F and the temperature ,as well as with the Legendre transform of the internal energyF, with respect to F and the specific entropy .

The result derived here generalizes an earlier one which hadbeen established for non-adiabatic processes 9,10,5. Thisformer characterization of driving traction was not valid in adia-batic processes, and therefore did not, in particular, apply to shockwaves in classical gas dynamics or to impact-induced rapidlymoving phase boundaries in solids. A one-dimensional version of

the present result was obtained in 11.

2 Momentum and Energy

Consider a body which occupies a region R in a reference con-figuration. Let xR denote the position of a particle in this con-figuration and let t denote time. Consider a thermomechanicalprocess of this body on some time interval t1 ,t2 which is char-acterized by the motion y(x,t), body force per unit mass b(x,t) ,Piola-Kirchhoff stress (x,t), heat flux q(x,t), heat supply r(x,t)and internal energy per unit mass (x,t). Suppose that during thisprocess y is continuous with piecewise continuous first and sec-ond derivatives on R t1 ,t2; b( ,t) and r( ,t) are continuouson R; (,t) and q( ,t) are piecewise continuous with piecewisecontinuous gradient on R; and is piecewise continuous withpiecewise continuous first derivatives on Rt1 ,t2. During this

process, the usual balance laws of linear and angular momentumand the first law of thermodynamics require that for any subregion

D,

D

n dAD

b dVd

dt

D

v dV, (1)

D

yn dAD

yb dVd

dt

D

yv dV, (2)

D

nvqndAD

bvrdV

d

dt

D 1

2vv dV. (3)

Here vy denotes particle velocity, x is the mass density in thereference configuration which is assumed to be continuous on R,and n is a unit outward normal vector on D.

At a point in R at which the fields are smooth the balance laws13 yield the usual field equations

Div bv, (4)

FTFT, (5)

FDiv qr, (6)

where FGrad y is the deformation gradient tensor.

Next, suppose that there is a surface S t in R such that the fieldsF, v, q, and suffer jump discontinuities across S t while beingcontinuous on either side of it. Such a surface may represent, forexample, the Lagrangian image of a shock wave or an interfaceseparating two material phases. Let Vn0 denote the normal ve-locity of propagation of this interface. We refer to the side intowhich Vn points as the positive side of S t . For any field quantity

g(x,t) let g

and g

denote the limiting values of g as a point onS t is approached from its positive and negative side, respectively.Then, we let g and g denote the jump and the average valuesof g on S t :

g g g

, g1

2 g g. (7)

At a point on S t , the balance laws 13 yield the usual jumpconditions

nvVn0, (8)

nv 12vv Vn qn0. (9)

The energy jump condition 9 can be written in the followingalternative form by making use of 8 and vVnFn0which follows from the continuity of the deformation for alge-braic details see, for example, 5:

F Vn qn . (10)

3 Rate of Entropy Production

In order to address the second law of thermodynamics one mustconsider two additional fields, viz. the temperature (x,t) and theentropy per unit mass (x,t). Suppose that ( ,t) is piecewisecontinuous with a piecewise continuous gradient on R, and that is piecewise continuous with piecewise continuous first deriva-tives on R t1 ,t2; and are permitted to suffer jump discon-tinuities across S t . The rate of entropy production associated witha subregion D is defined by

d

dt

D

dVD

qn

dA

D

r

dV, (11)

and the second law of thermodynamics requires that 0 for allregions D and all processes. When the region D intersects theinterface S t one can rewrite 11 in the form

D Div q

r

dV

S tD

Vn qn dA (12)by carrying out a standard calculation; e.g. see page 116 of 12.The first term in 12 represents the entropy production rate in thebulk of the body and the second term is associated with the mov-ing interface. Let s denote the rate of entropy production due tothe propagating surface:

sS t

Vn qn dA . (13)One finds by using 13 and 10, that s can be alternativelyexpressed as

sS t

1 F Vnqn 1

dA . (14)

830 Vol. 67, DECEMBER 2000 Transactions of the ASME


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In an adiabatic process there is no heat transfer: q0 and r0. On the other hand if the process is not adiabatic, the typicalheat conduction law, whatever it may be, involves Grad andtherefore the partial differential equations resulting from using theconstitutive relationships in the energy Eq. 6 involve at leastthe second spatial derivative of ; thus, one usually requires thetemperature to be continuous in non-adiabatic processes: 0on S t . Thus in both the adiabatic and non-adiabatic cases one has

q

0 on S t and therefore necessarily

q

n

1

0 and 1

1

q

n

0. (15)

In view of this and 10, we can write s as

sS t

F

Vn dA , (16)

or in terms of the Helmholtz free-energy as

sS t

F

Vn dA . (17)

4 Driving Traction

The rate of entropy production due to the propagating interfacecan be written as

sS tf Vn

dA (18)

where

fF

F (19)

is called the driving traction or Eshelby force. The second law ofthermodynamics requires that f Vn0 on S t which specifies thedirection in which the interface is permitted to move. This result isvalid for any continuum undergoing an arbitrary thermomechani-cal process which may or may not be adiabatic. If the process isadiabatic, 19 and 10 yield f . If it is not adia-batic, 19 specializes to f F .

In the special case of a thermoelastic material one has F, and the stress and entropy are given by the constitutive

relationships F , . Equivalently one has F,

with the stress and temperature given by F , . Thusfor a thermoelastic material 19 can be written as

f FF

FF (20)

which is reminiscent of the Legendre transforms of F, andF,.

Acknowledgment

The results reported here were obtained in the course of re-search supported by the National Science Foundation.

References

1 Eshelby, J. D., 1956, Continuum Theory of Lattice Defects, Solid StatePhysics, Vol. 3, F. Seitz and D. Turnbull, eds., Academic Press, San Diego, pp.

79144.

2 Callen, H. B., 1985, Thermodynamics and an Introduction to Thermostatistics,Second Ed., John Wiley and Sons, New York, Chapter 14.

3 Kestin, J., 1968, A Course on Thermodynamics, Vol. II, McGraw-Hill, NewYork, Chapter 14.

4 Truesdell, C., 1969, Rational Thermodynamics Lecture 7, Springer-Verlag,New York.

5 Abeyaratne, R., and Knowles, J. K., 1990, On the Driving Traction Acting ona Surface of Strain Discontinuity in a Continuum, J. Mech. Phys. Solids, 38,

pp. 345360.

6 Abeyaratne, R., and Knowles, J. K., 1991, Kinetic Relations and the Propa-gation of Phase Boundaries in Solids, Arch. Ration. Mech. Anal., 114, pp.

119154.7 Abeyaratne, R., Kim, S-J., and Knowles, J. K., 1994, A One-Dimensional

Continuum Model for Shape-Memory Alloys, Int. J. Solids Struct., 31, pp.

22292249.8 Rosakis, P., and Tsai, H., 1995, Dynamic Twinning Processes in Crystals,

Int. J. Solids Struct., 32, pp. 27112723.

9 Heidug, W., and Lehner, F. K., 1985, Thermodynamics of Coherent PhaseTransformations in Non-Hydrostatically Stressed Solids, Pure Appl. Geo-

phys., 123, pp. 9198.10 Truskinovsky, L., 1985, Structure of an Isothermal Phase Discontinuity,

Sov. Phys. Dokl., 30, pp. 945948.

11 Abeyaratne, R., and Knowles, J. K., 1994, Dynamics of Propagating Phase

Boundaries: Adiabatic Theory for Thermoelastic Solids, Physica D, 79, pp.269288.12 Chadwick, P., 1976, Continuum Mechanics, John Wiley and Sons, New York.

Characterizing Damping and

Restitution in Compliant Impacts

via Modified K-V and Higher-Order

Linear Viscoelastic Models

E. A. ButcherAssoc. Mem. ASME, Department

of Mechanical Engineering,

University of Alaska, Fairbanks, AK 99775-5905

D. J. SegalmanFellow ASME, Sandia National Laboratories,1

P.O. Box 5800,

MS 0847, Albuquerque, NM 87185-0847

1 Introduction

Time-domain models for compliant impacts have been widelyused to model collision dynamics as finite-time events. The most

common way to account for energy dissipation in the compliantimpact model has been via the standard Kelvin-Voigt K-V vis-coelastic model

F tkxcx (1)

in which the resulting equation of motion assumes the familiarlinear form

x2nxn2x0 (2)

from vibration theory where nk/m and c/(2km ). Theinitial conditions x(0 )0 and x(0 )v0 yield the solution

x tv0

dexpntsin dt (3)

where dn12. If the impact duration is assumed to be a

half-period of vibration associated with the damped frequency,then the exact restitution coefficient is obtained easily in terms ofthe dimensionless damping ratio as

1Sandia National Laboratories is a multiprogram laboratory operated by Sandia

Corporation, A Lockheed Martin Company for the U.S. Department of Energy under

Contract DE-ACO4, 94AL85000.



MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Au-

gust 25, 1998; final revision, February 10, 2000. Associate Technical Editor: V. K.

Kinra.



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ex tr

x0 exp 12 (4)

where tr/d is the release time 1,2. While the undampedcollision is elastic, for critical damping or overdamping the colli-sion is purely plastic. Another reason the half-period K-V modelhas been widely used is that Eq. 4 may easily be used to obtainthe impact damping parameter c as

c2 ln e km ln e22

(5)

in terms of an experimentally obtained restitution coefficient 3.The energy EL lost in the impact is

ELE01e2E0 1exp 212 (6)

where E0 is the initial kinetic energy. This energy loss is repre-sented in Fig. 1 by the area enclosed inside the hysteresis curveO-A-B-D-O. The peak elastic potential energy stored in the im-pact is

U1

2kx max

2E0 exp 212 tan1

12

(7)from which an equivalent linear damping ratio eq may be foundvia the loss factor EL /(2U) as

eq

2

1

4

exp

2

12

tan112

1exp 212 (8)

Thus the bilinear impact model may be replaced by an equivalentlinear Kelvin-Voigt model with damping constant c eq2eqkm which dissipates EL energy per period of vibration.This technique is often advantageous in vibratory impact prob-lems.

Hunt and Crossley 4 noticed, however, that linear viscousdamping in the K-V model gives an unrealistic hysteresis diagramfor the impact force-deflection curve. Specifically, they noted that

the parallel linear dashpot results in discontinuous force profiles atinitial contact and release as well as a nonphysical tensile forceapplied during the end of the restitution phase. In order to elimi-nate this force discontinuity, they suggested a nonlinear dampingfunction for use with the Hertzian stiffness model which complieswith the expected boundary conditions of vanishing force atcontact and release. Estimates of the corresponding restitutioncoefficient in terms of the models damping parameters weremade by the above authors and Herbert and McWhannell 5,who also noted that the effects of eliminating the force disconti-nuities include a more realistic frequency content in the impulse

generated.While other authors e.g., 6,7 have proposed different non-linear models which also satisfy the expected boundary condi-tions, there have been few efforts to eliminate the force disconti-nuities in the K-V model while remaining within the frameworkof linear viscoelasticity. This paper attempts to help fill this gap.In contrast to using the various nonlinear models, this approachenables the associated damping and restitution to be characterizedanalytically without the need to approximate. First, the standardK-V model is reconsidered here under a different assumption re-garding the restitution phase: that the mass releases when the

force vanishesbefore the initial contact location is reached. Thisapproach was recently used by Luo and Hanagud 8 in order toimprove the modeling and simulation of vibration absorbers withmotion-limiting stops. In order also to guarantee that the forcevanishes upon impact, higher-order viscoelastic Maxwell andstandard linear models are implemented in which the boundaryconditions in the force-displacement hysteresis curve are all satis-fied and the force history is entirely continuous. A similar modelfor the impact surface has been utilized in a previous paper 9in an effort to circumvent the previously mentioned discontinui-ties in the dynamic model of robotic manipulator collisions. Un-like these studies which were concerned with simulation, how-ever, this paper presents analytical values of the restitutioncoefficient and related quantities for the viscoelastic models interms of the dimensionless viscoelastic parameters. Hysteresisdiagrams and restitution coefficients for each model are plottedand compared.

2 Modified K-V ModelIn order to eliminate the tension in the K-V model, a better

representation of the dynamics allows the mass to release whenthe net force vanishes. In Fig. 1, this occurs at point C. Theresulting area of the hysteresis curve O-A-B-C-O the energyloss is thus smaller than that obtained using a half-period ofvibration. Setting the force in Eq. 1 to zero, the release time isfound as

tr1

dtan1 212

221 (9)

which yields the restitution coefficient

eexp

12

tan1

212

2

2

1 ; 1 (10)

As 1,eexp(2)0.14 so that, unlike the half-period ver-sion, the impact is not perfectly plastic when the damping is criti-cal. Instead, the nonzero restitution coefficient

e 2121

/21

; 1 (11)

which matches Eq. 10 for 1 is obtained for overdamping. The

energy lost in an impact is ELE0(1e2). Since the peak

potential energy is given by Eq. 7 for 1 and by UE0e for1, the equivalent linear damping ratio may be obtained as

Fig. 1 Hysteresis diagrams for the Kelvin-Voigt solid, Max-well dotted, and standard linear impact models with n1and 0.1 where 0.0 solid, 0.05 long-dashed, 0.2 short-dashed, and 0.4 short-long-dashed. The modified K-V andstandard linear models omit the tension at the conclusion ofthe restitution phase of impact.



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eq1

4exp 212 tan1

12

1exp 212 tan1212

221 ; 1

1

2sinh 21 ln

21

21 ; 1

(12)

from which an equivalent damping constant for use in vibratoryimpact is ceq2eqkm . Although eq is less than in the half-period model, the difference remains less than 0.02 for 1. Thus,eliminating the force discontinuity at release results in a minimaldecrease in equivalent damping.

3 Maxwell Model

The force discontinuity at impact cannot be eliminated inthe K-V model due to the models lack of an instantaneous elas-ticity. Instead, a higher-order viscoelastic model which portraysinstantaneous elasticity may be utilized. The most basic of these isthe Maxwell model in which the force-displacement relation is10

Fc

k

F cx . (13)

Equation 13 results in the third-order differential equation

x2nxn2x0 (14)

describing the impact dynamics where nk/m and km/(2c). The initial conditions x(0)0, x(0 )v0 , and

x(0)0 yield the solution

x texpnt v0d 122sin dt2v0

ncos dt

2v0

n(15)

where dn12.

As seen in Fig. 1 hysteresis curve O-E-F-O, the discontinuityupon impact has been eliminated via the third initial condition.Furthermore, if release occurs when the force vanishes then allboundary conditions are satisfied. Since the release time is equiva-lent to a half-period ( tr/d), the restitution coefficient e andenergy lost EL are thus found to be equivalent to those obtained inthe half-period K-V model. Thus as 1,e0 and the collisionbecomes perfectly plastic. Since Eq. 4 also applies for the Max-well model, the damping parameter is easily calculated in terms ofthe coefficient of restitution as in the K-V model as

ckm ln e 22

2 ln e. (16)

Because the spring and dashpot are in series, the peak elasticpotential energy stored in the spring is found in terms of the

maximum force as

U1

2k Fmax

k

2

E0 exp 212 tan112

. (17)The equivalent linear damping ratio, therefore, is also equivalentto that for the half-period K-V model so that the Maxwell impactmodel may be replaced by a linear K-V model with c eq2eqkm which dissipates EL energy per period of vibration. Itshould be observed that, although certain quantities of the half-period K-V and Maxwell impact models are conveniently equiva-lent, their inherent physics are completely different as representedby the corresponding hysteresis curves.

4 Standard Linear Model

Another instantaneously elastic higher-order viscoelastic modelwhich can be utilized is the standard linear model which consistsof a K-V element with spring constant k2 in series with anotherspring k1 . The force-displacement relation is 10

k1k2FcFk1k2xk1cx . (18)

Equation 18 results in the third-order equation

2

nxx2nxn

2x0 (19)

for the impact dynamics where nk1k2 /(( k1k2)m), whilek1c/(2(k1k2)mn) and k2 /(k1k2) are the dimension-less viscoelastic parameters. In the limit as k1, then nk2 /m ,2nc/m,0, and the system becomes a K-Vmodel with natural frequency n and damping ratio . Hence for1, Eq. 19 represents a perturbation of the standard K-Vmodel Eq. 2. Although the exact solution and restitution coef-ficient are intractable in this model, an approximate closed-formsolution may be obtained be means of a singular perturbationtechnique 11 in which the three roots and i are ob-tained to first order in as

n

22n14

2n

n142n

n12342

2

12n . (20)

The perturbation series converges providing 1/(2) .The force-displacement hysteresis curves in Fig. 1 correspond

to different values of the dimensionless parameter . This param-eter affects the models instantaneous elasticity and can be ad-

justed to sufficiently smooth out the K-V force discontinuity atthe origin. By allowing the mass to release at vanishing force,each of the force boundary conditions remain satisfied. Hence, forsmall values of, the damping and restitution for this model areperturbations of those for the modified K-V model. The restitutioncoefficient was found for 1 to first order in as

eexp 12f1 tan1212

221f2

(21)

where f1() and f2() were found to expand as f1()3/2

O(5) and f2()233O(5). These approximations are

accurate for small and break down for near unity. In order toverify the analytical expression in Eq. 21 using the expansionsfor f1() and f2(), the restitution for different values was alsoobtained numerically from the final velocity at release. It wasfound that the two results are practically identical for small valuesof both and . Finally, unlike the previous models considered,an equivalent linear damping ratio is not easily obtained for thismodel in terms of and .

5 Discussion

The restitution coefficients for each of the viscoelastic impactmodels are plotted as a function of in Fig. 2 in which the half-

Journal of Applied Mechanics DECEMBER 2000, Vol. 67 833


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period K-V and Maxwell results are identical. The numerical re-sults for the standard linear model are shown for several values of. The corresponding perturbation results not shown are essen-tially the same for small and . It is seen that the restitutioncoefficient of the half-period K-V and Maxwell models indeedvanishes as 1. Hence, a finite damping constant cp may beassociated with purely plastic impacts in these two models wherecp2km for K-V and cpkm/2 for Maxwell. In the modifiedK-V model, however, the restitution vanishes only as so thatthe impact can never be purely plastic. It is also seen that themodified K-V model has a restitution coefficient which is veryclose to that of the half-period K-V and Maxwell models for smalldamping ratios since the release times in these models for lowdamping are nearly the same. The advantages of analytically ob-taining both the impact damping parameter in terms of an experi-mentally obtained restitution coefficient and the equivalent lineardamping constant for use in vibroimpact, together with the even

more significant fact that all of the force boundary conditions aresatisfied, leads to the conclusion that the Maxwell model is anattractive choice for practical implementation in the modeling ofdissipative compliant impacts. However, the standard linearmodel may also be helpful in smoothing out the K-V impactdiscontinuity. Especially if the impact duration is relatively longor there are additional static forces on the impact surface, thefinite static deformation of this model is preferable to the fluid-like behavior of the Maxwell model. Furthermore, if the dampingis small and the instantaneous stiffness is large, then the impactdynamics and restitution may be found as a perturbation of thosefor the modified K-V model as was done here. Finally, furtherwork is needed to extend these results to general planar and three-dimensional collision theories and to include the use of kineticand energetic restitution coefficients.

References

1 Yigit, A. S., Ulsoy, A. G., and Scott, R. A., 1990, Dynamics of a RadiallyRotating Beam With Impact, Part 1: Theoretical and Computational Model,

ASME J. Vibr. Acoust., 112, pp. 6570.

2 Brach, R. M., 1991, Mechanical Impact Dynamics: Rigid Body Collisions,John Wiley and Sons, New York.

3 Brogliato, B., 1996, Nonsmooth Impact Mechanics: Models, Dynamics, andControl, Springer-Verlag, New York.

4 Hunt, K. H., and Crossley, F. R. E., 1975, Coefficient of RestitutionInterpreted as Damping in Vibroimpact, ASME J. Appl. Mech., 42, pp.

440445.5 Herbert, R. G., and McWhannell, D. C., 1977, Shape and Frequency

Composition of Pulses From an Impact Pair, J. Eng. Ind., 99, p p.513518.

6 Lee, T. W., and Wang, A. C., 1983, On the Dynamics of Intermittent-MotionMechanisms, ASME J. Mech. Des., 105, pp. 534540.

7 Khulief, Y. A., and Shabana, A. A., 1987, A Continuous Force Model for theImpact Analysis of Flexible Multibody Systems, Mech. Mach. Theory, 22,

No. 3, pp. 213224.8 Luo, H., and Hanagud, S., 1998, On the Dynamics of Vibration Absorbers

With Motion-Limiting Stops, ASME J. Appl. Mech., 65, pp. 223233.

9 Mills, J. K., and Nguyen, C. V., 1992, Robotic Manipulator Collisions:Modeling and Simulation, ASME J. Dyn. Syst., Meas., Control, 114, pp.

650659.10 Haddad, Y. M., 1995, Viscoelasticity of Engineering Materials, Chapman and

Hall, New York.11 Nayfeh, A. H., 1981, Introduction to Perturbation Techniques, John Wiley and

Sons, New York.

Finite-Amplitude Elastic Instability of

Plane-Poiseuille Flow of

Viscoelastic Fluids

R. E. Khayate-mail: [email protected]

N. Ashrafi

Department of Mechanical and Materials Engineering,University of Western Ontario, London,

Ontario N6A 5B9, Canada

The purely elastic stability and bifurcation of the one-dimensionalplane Poiseuille flow is determined for a large class of Oldroydfluids with added viscosity, which typically represent polymer so-lutions composed of a Newtonian solvent and a polymeric solute.The problem is reduced to a nonlinear dynamical system using theGalerkin projection method. It is shown that elastic normal stresseffects can be solely responsible for the destabilization of the base(Poiseuille) flow. It is found that the stability and bifurcation pic-ture is dramatically influenced by the solvent-to-solute viscosity

ratio, . As the flow deviates from the Newtonian limit and decreases below a critical value, the base flow loses its stability.Two static bifurcations emerge at two critical Weissenberg num-bers, forming a closed diagram that widens as the level of elas-ticity increases. S0021-89360000703-0

1 Introduction

While the problem of stability of plane-Poiseuille flow PPFhas been extensively investigated for Newtonian fluids, relativelylittle attention has been devoted to the flow of viscoelastic fluids.The presence of viscoelasticity is expected to dramatically alterthe stability and bifurcation picture in PPF, and yet no study hasso far predicted the nonlinear bifurcation from the base flow. Thepresence of additional nonlinearities that are usually part of any

realistic constitutive model 1 are expected to lead to the depar-ture from the Newtonian picture. Similarly to the case of Taylor-Couette flow, there is experimental evidence that the base flow ina channel may lose its stability as a result of fluid elasticity insidethe tube 2. This mechanism is now known as constitutive in-stability, as opposed to stick-slip induced instability. This mecha-nism of loss of stability should not be confounded with the short-wave instability due to a change in type of the field equations,



MECHANICS. Manuscript received by the ASME Applied Mechanics Division, April

28, 1999; final revision, July 27, 1999. Associate Technical Editor: A. K. Mal.

Fig. 2 Restitution coefficients for the half-period K-V and Max-well models dotted, modified K-V solid, and standard linearimpact model numerical results where 0.0 solid, 0.05long-dashed, 0.2 short-dashed, and 0.4 short-long-dashed

834 Vol. 67, DECEMBER 2000 Copyright 2000 by ASME Transactions of the ASME


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which is known as Hadamard instability 3. The emergence ofsurface instability at the exit of an extrusion die sharkskin andmelt fracture keeps hinting at the possibility of a link with ahydrodynamic instability inside the channel, away and upstreamfrom the exit 4,5. However, linear stability analyses of channelCouette and Poiseuille flows, using elementary constitutivemodels, such as Maxwell and Oldroyd-B fluids, failed to assertthat the base flow may be linearly unstable when the level ofelasticity Weissenberg number exceeds a critical level 4,5.More recent studies based on more generalized constitutive mod-els of the Oldroyd class showed that the base flow in a channel

can become unstable to small perturbations for some range ofWeissenberg numbers 6 8. These generalized constitutivemodels display a nonmonotonic shear-stress/shear-rate curve. Therange of instability coincides with the negative slope of the stresscurve. However, only linear stability analyses were carried out.

The present study focuses on the nonlinear constitutive insta-bility of the PPF of high-molecular-weight fluids. These fluids aretypically composed of a Newtonian solvent and a polymeric sol-ute. The Johnson-Segalman JS constitutive model is used, whichis a highly nonlinear equation, and is one of the very few consti-tutive models that exhibit a nonmonotonic stress/shear-rate curve.It is thus expected that, while the presence of inertia and shearthinning alone can destabilize the flow, fluid elasticity or normalstresses will give rise to additional nonlinearities and couplingamong the flow variables, making an already complex problem

due mainly to inertia even more difficult to solve. Similarly toany flow in the transition regime, the PPF of viscoelastic fluidsinvolves a continuous range of excited spatio-temporal scales. Inorder to assess the influence of the arbitrarily many smaller lengthand time scales on the flow, one would have to resort to theresolution of the flow at the small-scale level. This issue remainsunresolved since, despite the great advances in storage and speedof modern computers, it will not be possible to resolve all of thecontinuous ranges of scales in the transition regime.

It is by now well established that dynamical systems can be aviable alternative to conventional numerical methods as oneprobes the nonlinear range of flow behavior 9. Dynamical sys-tems are obtained using the Galerkin approximation. The velocityand stress components assume truncated Fourier or other orthogo-nal representations in space, depending on the boundary condi-tions. The expansion coefficients are functions of time alone, thus

leading to a nonlinear system upon projection of the equationsonto the various modes. The relative simplicity of dynamical sys-tems, and the rich sequence of nonlinear flow phenomena exhib-ited by their solution, have been the major contributing factors totheir widespread use as models for examining the onset of non-linear behavior. The dynamical system approach has typicallybeen used to handle simple flow configurations, and most particu-larly Newtonian flows. Recently, this approach has been at-tempted for non-Newtonian flows in thermal convection 1012 and rotating flow 1317. For Taylor-Couette flow,comparison was carried out with the experiments of Muller et al.18, leading to excellent agreement 15. A modal expansionsimilar to that in 14,15 is used to solve the current problem.

2 Problem Formulation and Solution Procedure

Consider the plane channel Poiseuille flow of an incompress-ible viscoelastic fluid of density , relaxation time , and viscosity. In this study, only fluids that can be reasonably represented bya single relaxation time and constant viscosity are considered. Thefluid considered here, is a polymer solution composed of a New-tonian solvent and a polymer solute of viscosities s and p ,respectively. Therefore sp . The velocity, time, space co-ordinates, pressure, and stress are nondimensionalized by d/, ,d, pU/d and p /, respectively. Here Uis the maximum veloc-ity of the base Poiseuille flow, and d is the gap between the twoplates. There are three important similarity groups in the problem,

namely, the Reynolds number, Re, the Weissenberg number, We,and the solvent-to-solute viscosity ratio, , which are given, re-spectively, by

Red2

p, We

U

d,

s

p

. (1)

The continuity and conservation of momentum equations for ageneral incompressible viscoelastic fluid are given in dimension-less form as

u0, Redu

dtWep 2u (2)

where u is the velocity vector, p is the pressure, is the polymericcontribution of the stress tensor, t is the time, d/dt is the substan-tial derivative operator, and is the gradient operator. The con-stitutive equation adopted in this study belongs to the Oldroydclass of incompressible viscoelastic fluids:

d

dt 1

2 utu

2uut

uu t (3)

where (u) t denotes the transpose ofu. Equation 3 includesboth lower and upper-convective terms. It is often referred to asthe Johnson-Segalman model 19. Here 0,2, which is adimensionless material slip parameter. The value of is a mea-sure of the contribution of nonaffine motion to the shear tensor.For 0, the motion is affine and the Oldroyd-B model is recov-ered, whereas for 2, the motion is completely nonaffine andthe model is reduced to the Oldroyd-Jaumann model 4. When0 and s0, the upper-convected Maxwell model isrecovered.

If the x-axis is taken to lie halfway between the two plates, andy is the coordinate in the transverse direction, then the total shearstress corresponding to the base Poiseuille flow is given by

Txyb

12 2We y (4)

where du/dy is the shear rate and u is the velocity in thex-direction. Note that We is the dimensionless driving pressuregradient. Equation 4 is perhaps the most revealing result of theJS model. It reflects the possibility of a nonmonotonic behaviorfor the stress/shear-rate relation. Indeed, Fig. 1 shows the behavior

of the shear stress, Txyb

, as a function of for 0,1 and

Fig. 1 Steady-state shear stress versus shear-rate curves for0.2 and 0,1. The loci of the two extrema are also shown,which join into one curve denoted here by c . The curves inthe figure resemble the pressurestretch-ratio related to the in-flation of a Mooney-Rivlin material see Fig. 2 in 20.



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0.2. The curve 1 corresponds essentially to the Newtonian

limit. For this value of the viscosity ratio, the elastic contributionto Txy

bin expression 4 is negligible. In this case Newtons law of

viscosity applies. The figure indicates that the stress curves losemonotonicity for 1/8. Two extrema a maximum and a mini-mum appear, which tend to merge as increases, as indicated bythe curve joining the loci of the extrema. The base flow is found tobe unstable for the range of the stress curves with negative slope.This situation is reminiscent of the load/deformation behavior inelasticity. In the case of nonlinear inflation of a Mooney-Rivlinhyperelastic membrane, for instance, the pressure also exhibits asimilar behavior as function of the stretch ratio for variousMooney constants 20. Upon comparison with the curves inFig. 1, the curve 0 is comparable to that of a Neo-Hookeansolid, while the curve for a Newtonian fluid (1) is comparableto the curve of a Hookean solid see Fig. 2 in 20.

The solution of the system 12 is carried out using theGalerkin projection method. For one-dimensional disturbancealong the channel x-axis, the departure from base flow is re-duced to the axial velocity, u(y ,t), normal stress difference,

N(y ,t), and shear stress, S(y ,t). In this case, Eqs. 13 reduceto

Re u tuyySy (5a)

NtN2We SSuySbuy . (5b)

S tSuyaWeNNuyNbuy (5c)

where a(/21). Here S b/1(2) 2 is the non-Newtonian contribution of the shear stress of the base flow, and

Nb2 2/1(2) 2 is the corresponding first normal stressdifference. Note that a subscript in Eqs. 5 denotes partial differ-

entiation. The flow departure is represented by series of Chan-drasekhar functions, which satisfy the homogeneous no-slipboundary conditions 15. A judicious selection process andtruncation level is applied for the choice of the various modes inorder to ensure the physical and mathematical coherence of thefinal model.

3 Bifurcation and Stability Picture

While the linear stability picture is somewhat predictable, thebifurcation picture is far from being intuitively obvious. The bi-furcation diagrams depend strongly on and . We thus monitorthe influence of the viscosity ratio by fixing the parameter to 0.2

and varying . Figure 2 displays the resulting bifurcation diagramsin the N, We plane for 0.06, 0.08 . The figure shows thedependence of the steady-state normal stress difference, N(0,) ,at the center of the channel. Linear stability analysis asserts that,

for large value 1/8, the base flow is stable to small pertur-bations for any value of We. This situation corresponds to amonotonic shear-stress/shear-rate curve in Fig. 1. As decreases,two extrema appear in the stress curves in Fig. 1, entraining a loss

of stability of the base flow in between. For each 1/8, a closedbifurcation diagram emerges as depicted in Fig. 2, which shows a

widening of the unstable range of We values as the viscosity ratio

decreases.Although the case 0.06 will be discussed in detail below, we

examine first the evolution of the bifurcation and stability picture

as the flow deviates from close to the Newtonian limit this limitis approached when the solvent-to-solute viscosity ratio, , ishigh. As decreases below a critical value, two static bifurca-tions emerge at two critical values, Wec1 and Wec2 , of the Weis-

senberg number as predicted by linear stability analysis. The two

critical points coincide with points A and E for the 0.06 dia-gram. The two bifurcating branches join over the unstable range

to form a closed diagram. This is clearly illustrated for 0.08;the closed diagram intersects the We axis at Wec18.48 and

Wec 216.53. As decreases further, the closed diagram wid-ens, and another closed diagram appears as depicted in Fig. 2 for

0.076. In this case, there are four critical values of the Weis-

senberg number that are present at 8.4, 17, 30, and 34.5. Thesecond range of We values 30 to 34.5 corresponds to unstablebase flow. A stable range exists between the two diagrams. As decreases further, the two diagrams grow, come in contact with

one another, and finally merge to form a simply connected closed

diagram as shown in Fig. 2 for 0.06. In this case, the range ofinstability of the base flow becomes larger as it covers the values

7.54We48.5 between A and E. The figure also indicates thatthe solution branch changes concavity, and presents regions alter-

nating in stability.

In general, and as typically depicted by the 0.06 diagram,there is an exchange of stability at the two critical points Wec 1 Aand Wec2 E, with the base flow losing its stability at Wec1 andregaining it at Wec2 . However, the base flow is not always un-

conditionally stable for WeWec1 and WeWec2 ; simulta-

neously, the diagram 0.06 is not always unconditionally stablefor Wec 1WeWec 2 . We have indicated in Fig. 2 the various

branches of alternating stability of the 0.06 diagram. Thus,branches AB, CD, EF, and GH are unstable, while the branches

BC, DE, FG, and HA are stable. Consequently, close to each

critical point, just before Wec1 and just after Wec2 , there is a

branch, BC and FG, respectively, to which the flow can converge

if the perturbation is not small, similarly to what occurs in the

vicinity of transcritical and subcritical bifurcations. Although

there are stable and unstable nontrivial branches in the range

Wec 1WeWec 2 , there is total loss of stability of the base flow.

In this range, only nonlinear velocity profiles are stable. The sta-

bility of the branches at the two critical points was established

numerically since linear stability analysis cannot be applied in the

vicinity of the critical nonhyperbolic fixed points.It is, perhaps, at this stage that one begins to connect the sta-

bility and bifurcation picture to physical reality. It is well known

that in real systems, physical instabilities are observed when the

flow rate and/or the level of elasticity are high. These instabilities

are believed to be potentially responsible for the onset of surface

roughness in extrusion 10. If we note that the flow rate is con-trolled by We, and the level of elasticity controlled by both We

and , then we can clearly observe that the trend shown in Fig. 2confirms that both the flow rate and fluid elasticity are the deter-

mining factors behind the destabilization of the base flow. It is

also well known that instabilities are suspected to set in after the

Weissenberg number has reached a certain value. This is also

Fig. 2 Bifurcation diagrams for the normal stress difference,N0,, at the center of the channel as function of We for 0.2 and 0.06,0.08. The smallest diagram corresponds tothe highest viscosity ratio, . As exceeds a critical level inthis case 18, the closed diagram reduces to zero, as thebase flow is always stable. The branches AB, CD, EF, and GH ofdiagram 0.06 are unstable, whereas the branches BC, DE,EF, FG, and HA are stable.



10/20

inferred from Fig. 2, as the diagram corresponding to 0.06, forinstance, indicates that the base flow is practically unstable for thewhole range We7.5.

In summary, a nonlinear analysis is carried out to examine theonset of constitutive instability and bifurcation for polymer solu-tions. Only elastic effects, which lead to the well-known Weissen-berg rod-climbing phenomenon, are responsible for the loss ofstability of the base Poiseuille flow. The viscoelastic model useddisplays nonmonotonicity of the shear-stress/shear-rate curve, andbelongs to the wider class of Oldroyd constitutive models thatlead to the destabilization of channel flow. The bifurcation dia-

grams are obtained for the first time, and show how the secondaryflow evolves as one deviates from the Newtonian limit. The bi-furcation diagrams are always closed and widen in range as thesolvent-to-solute viscosity decreases, thus reflecting the destabili-zation observed in practice as the level of elasticity increases. It isemphasized that the present stability and bifurcation picture cor-responds to perturbations of infinite wavelength, which may notbe the most dangerous modes. Only a higher-dimensional stabilityanalysis can indicate whether the present findings are of physicalrelevance.

Acknowledgments

We would like thank J. M. Floryan UWO for helpful com-ments on the manuscript. This work is funded by the Natural

Sciences and Engineering Research Council of Canada.

References

1 Bird, R. B., Armstrong, R. C., and Hassager, O., 1987, Dynamics of PolymericLiquids, Vol. 1, 2nd Ed., John Wiley and Sons, New York.

2 Vinogradov, G. V., Malkin, A. Ya., Vanovskii, Yu G., Borisenkova, E. K.,Yarlykov, B. V., and Berezheneya, G. V., 1972, J. Polym. Sci., Part A: Gen.

Pap., 10, p. 1061.3 Joseph, D. D., Renardy, M., and Saut, J. C., 1985, Hyperbolicity and Change

of Type in the Flow of Viscoelastic Fluids, Arch. Ration. Mech. Anal., 87, p.213.

4 Denn, M. M., 1990, Issues in Viscoelastic Fluid Mechanics, Annu. Rev.Fluid Mech., 22, p. 13.

5 Larson, R. G., 1992, Instabilities in Viscoelastic Flows, Rheol. Acta, 31, p.213.

6 Kolkka, R. W., Malkus, D. S., Hansen, M. G., and Ierley, G. R., 1988, Spurt

Phenomena of the Johnson-Segalman Fluid and Related Models, J. Non-Newtonian Fluid Mech., 29, p. 303.

7 Malkus, D. S., Nohel, J. A., and Plohr, B. J., 1990, Dynamics of Shear Flowof a Non-Newtonian Fluid, J. Comput. Phys., 87, p. 464.

8 Georgiou, G. C., and Vlassopoulos, D., 1998, On the Stability of the SimpleShear Flow of a Johnson-Segalman Fluid, J. Non-Newtonian Fluid Mech.,75, p. 77.

9 Sell, G. R., Foias, C., and Temam, R., 1993, Turbulence in Fluid Flows: A Dynamical Systems Approach, Springer-Verlag, New York.

10 Khayat, R. E., 1994, Chaos and Overstability in the Thermal Convection ofViscoelastic Fluids, J. Non-Newtonian Fluid Mech., 53, p. 227.

11 Khayat, R. E., 1995, Nonlinear Overstability in the Thermal Convection ofViscoelastic Fluids, J. Non-Newtonian Fluid Mech., 58, p. 331.

12 Khayat, R. E., 1995, Fluid Elasticity and Transition of Chaos in ThermalConvection, Phys. Rev. E, 51, p. 380.

13 Avgousti, M., and Beris, A. N., 1993, Non-Axisymmetric Subcritical Bifur-cations in Viscoelastic Taylor-Couette Flow, Proc. R. Soc. London, Ser. A,A443, p. 17.

14 Khayat, R. E., 1995, Onset of Taylor Vortices and Chaos in Viscoelastic

Fluids, Phys. Fluids A, 7, p. 2191.15 Khayat, R. E., 1997, Low-Dimensional Approach to Nonlinear Overstabilityof Purely Elastic Taylor-Vortex Flow, Phys. Rev. Lett., 78, p. 4918.

16 Graham, M. D., 1998, Effect of Axial Flow on Viscoelastic Taylor-CouetteInstability, J. Fluid Mech., 360, p. 341.

17 Ashrafi, N., and Khayat, R. E., 2000, Finite Amplitude Taylor-Vortex Flowof Weakly Shear-Thinning Fluids, Phys. Rev. E, 61, p. 1455.

18 Muller, S. J., Shaqfeh, E. S. J., and Larson, R. G., 1993, Experimental Studyof the Onset of Oscillatory Instability in Viscoelastic Taylor-Couette Flow, J.

Non-Newtonian Fluid Mech., 46, p. 315.19 Johnson, M. W., and Segalman, D., 1977, A Model for Viscoelastic Fluid

Behavior Which Allows Non-Affine Deformation, J. Non-Newtonian Fluid

Mech., 2, p. 278.

20 Khayat, R. E., and Derdouri, A., 1994, Inflation of Hyperelastic CylindricalMembranes as Applied to Blow Moulding, Part I. Axisymmetric Case, Int. J.

Numer. Methods Eng., 37, p. 3773.

In-Plane Gravity Loading

of a Circular Membrane

R. O. TejedaResearch Assistant

E. G. LovellProfessor, Mem. ASME

R. L. EngelstadProfessor, Mem. ASME

Department of Mechanical Engineering, University of

Wisconsin-Madison, Madison, WI 53706-1572

This paper develops the displacement field for a circular mem-brane which is statically loaded by gravity acting in its plane.Coupled to the displacements are the stress and strain distribu-tions. The solution is applicable to the modeling of next genera-tion lithographic masks, ion-beam projection lithography masksin particular. S0021-89360000803-5

1 Introduction

In most engineering applications, the displacement, stress, andstrain fields induced by gravity are negligible. However, in nextgeneration nonoptical lithography masks used for semiconductordevice fabrication, it is critical to predict and compensate for dis-tortions which could potentially alter the quality of the microcir-cuit that is to be manufactured. Typically, the allowable error in alithographic mask is only a fraction of the microcircuits mini-mum feature size 1. Since ion-beam projection lithographyIPL is targeting the production of sub-100 nm devices, displace-ments due to gravity can be significant.

An IPL mask is composed of a circular membrane that is sup-ported by a relatively stiff frame and held in a vertical orientation

during exposure 2. It is typically made of silicon with a diam-eter on the order of 200 mm and 3.0-m thickness. If the mask ismodeled as a circular membrane that is constrained on its perim-eter by a rigid ring and subjected to in-plane gravitational loading

as shown in Fig. 1, it can be considered as a plane stress prob-lem and solved directly by traditional applied elasticity methods.To the best of the authors knowledge, a solution to this problemhas not been presented in the elasticity literature.

2 Solution Development

The position of an arbitrary point on the membrane is definedby the polar coordinates (r,) with the origin taken at the center.All translational displacement components are constrained at theoutside radius, R. In general, radial u and circumferential (v)

displacements which arise from the loading of the membrane arerelated to the radial strain (r), circumferential strain (), andshear strain (r) by strain-displacement equations, and to radialnormal stress (r), circumferential normal stress (), and shearstress (r) by Hookes law, i.e.,

ru

r

1

Er (1)



MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Apr.

28, 1999; final revision, May 5, 2000. Associate Technical Editor: R. C. Benson.



11/20

u

r

1

r

v

1

Er (2)

r1

r

u

v

r

v

r

21

Er . (3)

Here, E is Youngs modulus and is Poissons ratio. Equilibriumof the shaded element shown in Fig. 1 requires that

r

r

r

r

1

r

r

g sin 0 (4)

1

r

r

r

2

rrg cos 0 (5)

where and g are the material mass density and gravitationalacceleration, respectively.

A solution form which uses Airys stress function, , can beexpressed as

r1

r

r

1

r22

2Fr, (6)

2

r2Gr, (7)

r

r 1

r

(8)

where F(r,) and G(r,) are arbitrary functions. SubstitutingEqs. 68 into the equilibrium Eqs. 4 and 5 and subsequentintegration yields

Gr,gr sin G2r (9)

Fr,gr sin F2r

r

1

r G2r. (10)

To preclude singular stresses, and without loss of generality, letF2()G 2(r)0. Therefore,

r1

r

r

1

r22

2gr sin (11)

2

r2gr sin . (12)

General forms of satisfying compatibility were originallygiven by Michell 3 for problems described in polar coordinates.From Timoshenkos 4 summary of this,

a0 log rb0r2c 0r

2 log rd0r2a0

a1

2rsin

b 1r3a 1r

1b 1r log rcos

c1

2rcos

d1r3c 1r

1d1r log rsin

n2

a nrnb nr

n2a nr

nbnr

n2cos n

n2

c nrndnr

n2c nr

ndnr

n2sin n. (13)

A stress function which gives nonperiodic or singular stresses isnot admissible. Therefore, all the terms except those with coeffi-cients ofb 0 , b1 , d1 , a n , bn , c n , and dn will be dropped. So, forn2,3,4, . . .

rnsin n,cos n (14)

r ,rn2sin n,cos n (15)

rrn2cos n,sin n. (16)

On a circular edge, Eqs. 1416 describe normal and shearstresses which vary harmonically. They are not necessary to solve

the basic problem. Indeed, they may lead to displacements whichcannot be zero at the boundary rR. Thus, a solution may beconstructed using relevant s established by Michell, plus addi-tional terms associated with the gravitational body forces:

b0r2b 1r

3 cos d1r3 sin . (17)

By the definition established in Eq. 8,

r2b 1r sin 2d1 r cos . (18)

Due to the symmetric nature of the problem, r(r,/2)0.Therefore, b10. Similarly, Eqs. 11 and 12 show that

r2b02d1r sin gr sin (19)

2b06d1r sin gr sin . (20)

Because it is associated with a hydrostatic loading such as uni-form membrane prestress, b0 will also be discarded. This leavesone unknown, d1 , which can be identified by noting that the cir-cumferential normal strain is zero on the boundary, i.e., (R,)0. By using Eqs. 19, 20, and 2, this yields

d1g

2

1

3(21)

and the stresses can now be written in the form

r2g

3r sin (22)

2g

3 r sin (23)

rg1

3r cos . (24)

Employing Eqs. 13, boundary conditions, and symmetry con-ditions gives the following strains and displacements in the mem-brane:

rr,2g

E

12

3r sin (25)

r,0 (26)

Fig. 1 In-plane gravity loading of a circular membrane



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rr,2g

E

12

3r cos (27)

ug

E

12

3

r2R2sin (28)

v

g

E

12

3r2R2cos . (29)

The displacement component distributions given by Eqs. 28 and29 are illustrated in Figs. 2 and 3.

Transformation of the stress components into an x-y coordinatesystem yields

x2g

3y (30)

y2g

3y (31)

xyg1

3x; (32)

where x and y are Cartesian coordinates, x is the normal stress inthe x-direction, y is the normal stress in the y-direction, and xyis the in-plane shear stress. As required, these stresses satisfyequilibrium equations in x and y, and the associated displacementcomponents vanish on a circular boundary.

3 Conclusion

An elasticity solution for the in-plane gravitational loading of acircular membrane has been established. Stress, strain, and dis-

placement fields satisfy symmetry, boundary conditions, equilib-rium, compatibility for a linear elastic material, and aresingularity-free.

The solution is useful for predicting the distortion of masksused in next generation lithography processes. It is particularlyappropriate to modeling the ion-beam lithography mask, whichconsists of a large circular membrane supported by a frame orwafer ring while it is in a vertical orientation.

Acknowledgments

This research was supported by International SEMATECH, theSemiconductor Research Corporation SRC, and the NationalScience Foundation Graduate Research Fellowship Program.

References

1 Gross, G., 1997, Ion Projection Lithography: Next Generation Technol-ogy?, J. Vac. Sci. Technol., B15, No. 6, pp. 21362138.

2 Tejeda, R., Engelstad, R., Lovell, E., and Berry, I., 1998, Analysis, Design,and Optimization of Ion-Beam Lithography Masks, Proc. SPIE, Emerging

Lithographic Technologies II, 3331, pp. 621628.3 Michell, J. H., 1899, On the Direct Determination of Stress in an Elastic

Solid, With Application to the Theory of Plates, Proc. London Math. Soc.,31, pp. 100124.

4 Timoshenko, S. P., and Goodier, J. N., 1970, Theory of Elasticity, 3rd Ed.,McGraw-Hill, New York, pp. 132135.

Free Vibration of a Spinning Stepped

Timoshenko Beam

S. D. YuAssistant Professor, Mem. ASME, Department of

Mechanical Engineering, Ryerson Polytechnic University,

350 Victoria Street, Toronto, ON M5B 2K3, Canada

W. L. CleghornProfessor, Department of Mechanical and Industrial

Engineering, University of Toronto, Five Kings College

Road, Toronto, ON M5S 3G8, Canada

The finite element method is employed in this paper to investigate free-vibration problems of a spinning stepped Timoshenko beamconsisting of a series of uniform segments. Each uniform segmentis considered a substructure which may be modeled using beam

finite elements of uniform cross section. Assembly of global equa-tion of motion of the entire beam is achieved using Lagrangesmultiplier method. The natural frequencies and mode shapes aresubsequently reduced with the help of linear transformations to astandard eigenvalue problem for which a set of natural frequen-cies and mode shapes may be easily obtained. Numerical results

for an overhung stepped beam consisting of three uniform seg-

ments are obtained and presented as an illustrative example.S00021-89360100101-5

1 Introduction

In studying machine tool vibration during a turning cuttingprocess, it is often necessary to conduct free-vibration analysis of



MECHANICS. Manuscript received and accepted by the ASME Applied Mechanics

Division, Sept. 1, 1999; final revision, Apr. 10, 2000. Associate Technical Editor: N.

C. Perkins.

Fig. 2 Radial displacement contours. Range is g1

2R2E3.

Fig. 3 Circumferential displacement contours. Range is

g12R2E3.



13/20

spinning stepped shaft. Bauer 1 presented an analytical study ofa rotating uniform Euler-Bernoulli beam with various combina-tions of simple boundary conditions. Lee et al. 2 studied thefree vibration of a rotating Rayleigh shaft using the modal analy-sis approach and Galerkins method. Katz et al. 3 investigatedthe dynamic responses of a uniform rotating shaft subjected to amoving load in the axial direction using both Rayleigh and Ti-moshenko beam theories. Zu and Han 4 presented analyticalsolutions for free vibration of a spinning uniform Timoshenkobeam with all combinations of the three classical boundaryconditions.

In this paper, free vibration of a rotating stepped Timoshenkobeam is investigated using the finite element method. To enhancethe accuracy of the computed eigenvalues and mode shapes, athree-node beam element, which permits the use of quintic poly-nomials as the interpolation function for both lateral displace-ments and bending angles, is utilized. For each of field variable,six nodal quantitiesthe variable and its derivative with respectto the axial coordinate at all three nodes are introduced to theelement displacement vector. Two lateral deflections and twobending angles at each axial location need be defined in beamflexural vibration. Therefore, a three-node beam element has 24nodal variables.

Use of the finite element method makes it possible to reduce thefree-vibration problem of a spinning beam to a standard eigen-value problem for which all eigenvalues and eigenvectors may bedetermined simultaneously. This is one advantage over the use ofan analytical method in which eigenvalues are determined bysearching for the roots of a characteristic equation. As an illustra-tive example, natural frequencies of an overhung stepped are ob-tained for several different spin rates.

2 Mathematical Procedure

In this section, the equations of motion of a stepped beam ofcircular cross section, as shown in Fig. 1, are presented using aninertial coordinate system. Parameters defining each segment arelength, diameter, and axial coordinate of the left-most plane.

2.1 Governing Differential Equations for a Substructure.In modeling the stepped beam, each segment is considered a sub-

structure. Within a substructure, the equations of motion may bewritten as 3

L2 00 L2

2t22 0 L1

L1 0 t L0 0

0 L0

uxxuyy

0

0

0

0

(1)

where ux and uy are lateral displacements of the beam centroid inthe x and y directions, respectively; x and y are angles of rota-tion of the plane normal to the beam centroid, measured in the

xoz and yo z coordinate planes, respectively; the three operatormatrices are defined as

L2

A 0

0 I

, L1

0 0

0 I

,

(2)

L0 GA 2z2 GA zGA

zGAEI

2

z2 .

In Eq. 2, is the volume mass density of the beam material; Gis the shear modulus; E is the modulus of elasticity; A is thecross-sectional area; Iis the second moment of area; is the shearcorrection factor 0.9 for solid circular cross section, 4; is thespin rate.

Assume that a uniform segment k is modeled using Ne ,k (k

1,2, . . . ,Ns) three-node beam elements. Within each element,the displacement vector uek is related to the nodal displacementvector qek by

uekNeqek 0l e (3)

where is the local coordinate; Ne() is the shape function ma-trix. The equations of motion of a uniform segment in terms of thenodal coordinate vector may be easily derived using the minimumpotential energy principle or the Galerkin principle.

2.2 Assembly of Equations of Motion for the SteppedBeam. The global equations of motion for the entire beam maybe formulated by enforcing continuity conditions across each in-terface between two adjacent substructures. The procedure is il-

lustrated here for a single interface at node j with zzj . The fourdisplacement and four force continuity conditions may be writtenin terms of the nodal displacements as

ux

x

x

kx

uy

y

y

ky

zz

j

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0

uxxxkxuyyyky

zz

j

Cjuxxxkxuyyyky

zz

j

(4)

where and are the shear rigidity ratio and the bending rigidityratio, defined as

GA 1

GA 2,

EI1

EI2. (5)

The continuity conditions may be implemented into theglobal equations of motion using the Lagrange multiplier method5. The final equations of motion of a stepped beam may bewritten as

MqGqKq0 (6)

where the global mass, gyroscopic and stiffness matrices for theentire stepped beam may be formulated from the correspondingmatrices for substructures and constraint matrices Cj . Forexample, the global mass matrix for a three-segment steppedbeam is

Fig. 1 An overhung stepped shaft



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MM11

1 M121 0 0 0

M211 M22

1C1

TM11

2 C1 C1T

M122 M13

2 0

0 M212 C1 M22

2 M232 0

0 M312 M32

2 M332C2

TM11

3 C2 C2T

M123

0 0 0 M213 C2 M22

3

.2.3 Boundary Conditions. The global equations of motionof a stepped beam may be readily modified to account for bound-

ary conditions at the ends using the penalty approach. In thispaper, an overhung shaft clamped at one end and free at the otheris investigated.

3 Standard Eigenvalue Problem

One of the most important tasks in free-vibration analysisis to determine the natural frequencies and mode shapes describedby a system of homogeneous second-order differential equa-tions and boundary conditions at the ends of a beam. To uti-lize commercial routines for eigenanalysis, the second-ordersystem may be replaced by the following equivalent first-ordersystem

AxBx (7)

where coefficient matrices A and B may be obtained from matri-ces M, K, and G.

Natural frequencies and mode shapes of a spinning Timoshenkobeam may then be reduced to seeking and X satisfying

AX1

BX. (8)

4 Numerical Results

Values of the material and geometric properties of a three seg-ment stepped beam under investigation are given in Table 1.Table 2 provides values of natural frequencies of an overhung

three segment stepped beam shown in Fig. 1. Five different spinrates are considered. For the case of zero spin rate, natural fre-quencies have matched pairs. For a nonzero spin rate, there aretwo natural frequencies developed around each at-rest value. Oneis associated with the forward precession mode; the other is asso-ciated with the backward precession mode.

5 ConclusionsThis paper presents a finite element analysis of free vibration

of a spinning stepped Timoshenko beam. Through use of theLagrange multipliers method, both displacement continuity andforce equilibrium conditions are satisfied at the interfaces of

joining substructures. Because of the use of higher-orderbeam finite elements for each substructure, highly accurate naturalfrequencies of a stepped beam of any desired mode may beobtained.

References

1 Bauer, H. F., 1980, Vibration of a Rotating Uniform Beam, Part 1: Orienta-tion in the Axis of Rotation, J. Sound Vib., 72, No. 2, pp. 177189.

2 Lee, C. W., Katz, R., Ulsoy, A. G., and Scott, R. A., 1988, Modal Analysis

of a Distributed Parameter Rotating Beam, J. Sound Vib., 122, No. 1, pp.119130.3 Katz, R., Lee, C. W., Ulsoy, A. G., and Scott, R. A., 1988, Dynamic Re-

sponses of a Rotating Shaft Subjected to a Moving Load, J. Sound Vib., 122,

No. 1, pp. 131148.4 Zu, J. W., and Han, R. P. S., 1992, Natural Frequencies and Normal Modes

of a Spinning Timoshenko Beam With General Boundary Conditions, ASME

J. Appl. Mech., 59, pp. S197S203.5 Tabarrok, B., and Rimrott, F. P. J., 1994, Variational Methods and Comple-

mentary Formulations in Dynamics, Kluwer, Dordrecht, The Netherlands.

Penetration Experiments With

Limestone Targets and Ogive-Nose

Steel Projectiles

D. J. FrewAssoc. Mem. ASME, Waterways Experiment Station,

Vicksburg, MS 39180-6199

M. J. ForrestalFellow ASME, Sandia National Laboratories,

Albuquerque, NM 87185-0303

S. J. HanchakUniversity of Dayton Research Institute,

Dayton, OH 45469-0182

We conducted three sets of depth-of-penetration experiments withlimestone targets and 3.0 caliber-radius-head (CRH), ogive-nosesteel rod projectiles. The ogive-nose rod projectiles with length-to-diameter ratios of ten were machined from 4340 Rc 45 and Aer

Met 100 Rc 53 steel, round stock and had diameters and massesof 7.1 mm, 0.020 kg; 12.7 mm, 0.117 kg; and 25.4 mm, 0.931 kg.



MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Nov.

29, 1999; final revision, May 10, 2000. Associate Technical Editor: I. M. Daniel.

Table 1 Material and geometric properties of a stepped spin-ning beam

Table 2 The first six natural frequencies of an overhungstepped beam



15/20

Powder guns or a two-stage, light-gas gun launched the projec-tiles at normal impacts to striking velocities between 0.4 and 1.9km/s. In addition, we present an analytical penetration equationthat described the target resistance by its density and a strength

parameter determined from depth of penetration versus strikingvelocity data. S0021-89360100201-X

Introduction

Several authors have written review articles that discuss themany analytical, computational, and experimental methods usedto study the broad field of penetration mechanics 1 5. Theresponses of the projectiles and targets depend strongly on theproblem geometry, materials, and impact conditions. Becausemany penetration mechanisms are possible, experimental observa-tions usually precede and guide analytical or computational mod-els. For this study, post-test observations showed a conical entrycrater with a depth of two or more projectile diameters followedby a circular penetration channel or tunnel with nearly the projec-tile diameter. For recent penetration studies with 6061-T6511 alu-minum targets 6,7 we could obtain post-test radiographs of thepenetration channels. However, we could obtain post-test obser-vations of the limestone penetration channels only after the targetswere split with the techniques used by stone masons. The projec-tiles recovered from the targets had small mass losses caused by

abrasion, but the overall nose shapes before and after penetrationlooked very similar. We previously observed similar post-test, tar-get channels and abraded projectiles with our studies on concretetargets 8,9. However, the concrete targets abraded the nose tipsmore severely than the limestone targets.

Based on penetration data sets with three projectile scales, wepresent an analytical penetration equation that describes target re-sistance by a parameter determined from penetration depth versusstriking velocity data. Our limestone penetration equation is simi-lar to that previously derived for concrete targets 10. However,for limestone targets, we observed a noticeable decrease in thetarget resistance parameter as the projectile diameters increased.So for the limestone targets, the target resistance parameter isequal to a constant term plus a term that depends on the projectilediameter. We hypothesize that the penetration equation will be

reasonably accurate for larger scale projectiles, but data frommuch more expensive field tests must be obtained to confirm ourhypothesis.

The limestone targets were quarried and cut by the Elliot StoneCompany.1 In the rock mechanics literature, this particular lime-stone is often called Salem, Indiana, or Bedford limestone. Forthis study, we conducted unconfined compression tests and sometriaxial compression tests on samples cored from representativeblocks and from individual targets before the penetration tests.Material properties from our targets are nearly the same as thosereported by Fossum, Senseny, Pfeifle, and Mellegard 11.

As previously mentioned, our penetration equation contains atarget strength constant that is determined from penetration depthversus striking velocity data. While this methodology provides anaccurate and convenient engineering equation, the detailed re-

sponse mechanisms for the target are not modeled. The authorsare not aware of any rigorous target models for rock penetrationproblems, but Lagrangian computational models that use adaptivemeshing techniques have shown promise for brittle ceramic tar-gets 12,13. Detailed computational approaches that model tar-get responses also require a broad array of quasi-static and dy-namic material properties data. For limestone, examples of somematerials experiments and data include 1 quasi-static, triaxialcompression experiments 11, 2 split Hopkinson bar experi-ments 1416, 3 shock wave studies 17, 4 dynamic ten-sile failure with planar-impact techniques 18,19, and 5

compression-shear loading with plate impact experiments 20.Data from other experimental techniques may also be required fora careful target analysis.

In the next sections, we present the penetration model, describethe experiments, and present our results and conclusions.

Penetration Model

For both limestone and concrete targets, post-test observationsshowed a conical entry crater with a depth of two or more projec-tile diameters followed by a circular channel or tunnel with nearly

the projectile diameter. The limestone penetration equations aresimilar to the previously published concrete penetration equations.From Forrestal, Altman, Cargile, and Hanchak10, depth of pen-etration P for an ogive-nose projectile and a concrete target isgiven by

Pm

2a2Nln 1 NV1

2

R4a, P4a (1a)

N81

242, V1

2

mVs24a3R

m4a 3N(1b)

in which the ogive-nose rod projectile is described by mass m,shank radius a, and caliber-radius-head . The target is describedby density and the target strength constant R. The strength con-stant is determined from

RNVs2

1 4a3N

m exp 2a 2P4aN

m1 (2)

where Vs is striking velocity. For a set of experiments, we hold allparameters constant and vary striking velocity. From each experi-ment, we measure striking velocity Vs and penetration depth P, so

R can be determined from 2 for each experiment. We then takethe average value of R from the data set and compare the predic-tion from 1 with the measured values ofVs and P. For this studywith limestone targets, ogive-nose steel rod projectiles with 7.1,12.7, and 25.4-mm diameters and a length-to-diameter ratio of tenhave values ofR913, 787, and 693 MPa, respectively. Thus, thetarget resistance decreases as the projectile shank diameter in-creases. We found that for these limestone targets

RKk2ao/2a (3)

in which K and k are constants obtained from data fits, 2 ao is areference projectile diameter, and 2a is the projectile diameter.We show later that with K607 MPa, k86MPa, and 2ao25.4 mm, we accurately recover the measured values of R foreach of the three data sets.

In summary, the procedure used to calculate R from penetrationdepth data for a fixed projectile is the same for concrete or lime-stone targets. However, for limestone targets, R depends on theprojectile shank diameter. Thus, we can use the penetration equa-tions 1a and 1b for limestone when R is given by 3.

Experiments

We conducted three sets of penetration experiments a total of30 experiments with ogive-nose steel rod projectiles and lime-stone targets. All projectiles had a total length-to-diameter ratioof ten and 3.0 caliber-radius-head CRH nose shapes. Theshank diameters and masses for each of the three sets of experi-ments were 7.1 mm, 0.020 kg; 12.7 mm, 0.117 kg; and 25.4 mm,0.931 kg.

Limestone Targets. The limestone targets were quarried andcut by the Elliot Stone Company of Bedford, IN. We obtained thetargets in three batches from nearby sites. Nominal material prop-erties for the three target batches are given in Table 1 and showminimal variations among the batches. In addition, we conducted1Elliot Stone Company, 3326 Mitchell Road, Bedford, IN 47421.



16/20

several triaxial compression tests 21,22 on samples from thethree target batches and found the brittle to ductile transition con-fining pressure to be about 50 MPa.

The 12.7-mm-Diameter, 0.117 kg, 3.0 CRH Projectiles.Our first set of experiments was conducted with steel projec-

tiles machined from 4340 Rc 45 23 round stock. Figure 1shows the projectile geometry, and for this set of experiments2a12.7 mm, L106 mm, and l21 mm. The target impactsurface was 0.51 m square, and the target lengths and other dataare given in Table 2. The sides and bottom of the targets weresurrounded by 0.10-m-thick concrete placed between a steel form

and the limestone. Six unconfined compression tests were con-ducted with 51-mm-diameter, 108-mm-long samples cored from arepresentative limestone block from Batch 1 and the averagestrength was c f58 MPa.

A 32-mm-diameter powder gun launched the 0.117 kg projec-tiles to the striking velocities recorded in Table 2. An additionalexperiment was conducted at Vs1605 m/s, but the trajectory was

Fig. 1 Projectile geometry

Table 1 Nominal material properties for the limestone targets

Table 2 Penetration data for the 12.7-mm-diameter 4340 Rc 4445, 0.117 kg, 3.0CRH projectiles. For ptich and yaw: Ddown, Uup, Rright, Lleft.

Table 3 Penetration data for the 7.1-mm-diameter 4340 Rc 4446 or Aer Met 100 Rc 53shots 4-1847 and 4-1846, 0.0205 kg, 3.0 CRH projectiles. For pitch and yaw: Ddown,Uup, Rright, Lleft.

Table 4 Penetration data for the 25.4-mm-diameter 4340Rc 4546, 0.931 kg, 3.0 CRHprojectiles. For pitch and yaw: Ddown, Uup, Rright, Lleft.



17/20

curved and the projectile exited the side of the target at a depth ofabout 0.65 m. The projectiles were fitted with sabots and obtura-tors that separated from the projectiles before impact. Four laserdiode systems measured striking velocities and orthogonal radio-graphs measured pitch and yaw angles. The target resistance R

was calculated from 2 for each experiment and recorded inTable 2. The average target resistance parameter for this set of

experiments is R787 MPa.

The 7.1-mm-Diameter 0.020 kg, 3.0 CRH Projectiles. Our

second set of experiments was conducted with steel projectiles

machined from both 4340 Rc 45 and Aer Met 100 Rc 53 24round stock. Figure 1 shows the projectile geometry, and for this

set of experiments 2a7.11 mm, L59.3 mm, and l11.8 mm.

A 20-mm-powder gun launched the 0.020 kg projectiles to strik-

ing velocities of 1230 m/s. For the larger striking velocities re-

corded in Table 3, a two-stage 50/20 mm light-gas gun launchedthe projectiles. The same target geometries and ballistics measure-

ments as those described for the 12.7-mm-diameter 0.117 kg pro-

jectiles were used for this set of experiments.

For this second set of experiments, we performed unconfined

compression tests on two samples cored from the targets. The

compressive strengths listed in Table 3 are the average of two

unconfined compression tests conducted with 50-mm-diameter

108-mm-long cores. In addition, we conducted penetration ex-

periments from both Batch 1 and Batch 3 limestone targets to

compare results from the two batches. We show later negli-

gible differences in the penetration data from both batches. The

average target resistance parameter of this set of experiments is

R913 MPa.

We also conducted an experiment with a 4340 Rc 45 projectile

at Vs1649 m/s. That projectile severely bent and turned withinthe target. Table 3 shows two experiments conducted with Aer

Met 100 Rc 53 projectiles. Shot 4-1846 with a striking velocity of

1674 m/s had a nearly straight trajectory. We then conducted ex-

periments at Vs1749, 1826, and 1863 m/s with Aer Met 100 Rc53 projectiles and these projectiles severely bent and turned

within the targets. Piekutowski, Forrestal, Poormon, and Warren

7 discusses in detail the better performance of the Aer Met 100Rc 53 projectiles.

The 25.4-mm-diameter 0.931 kg, 3.0 CRH Projectiles. Our

third set of experiments was conducted with steel projectiles ma-

Fig. 2 Post-test photographs of the 25.4-mm-diameterprojectiles

Fig. 3 Data and model predictions for the limestone targets



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chined from 4340 Rc 45 round stock. Figure 1 shows the projec-

tile geometry, and for this set of experiments 2 a25.4 mm, L212 mm, and l42 mm. The target impact surface was 1.02-msquare and the target lengths are given in Table 4. The sides andbottom

compliant impact

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